Motion Planning n Problem n Given start state x S , goal state x G n - - PDF document
Motion Planning Pieter Abbeel UC Berkeley EECS Many images from Lavalle, Planning Algorithms Motion Planning n Problem n Given start state x S , goal state x G n Asked for: a sequence of control inputs that leads from start to goal
Pieter Abbeel UC Berkeley EECS Many images from Lavalle, Planning Algorithms
n Problem
n Given start state xS, goal state xG n Asked for: a sequence of control inputs that leads from
n Why tricky?
n Need to avoid obstacles n For systems with underactuated dynamics: can’t simply
n E.g., car, helicopter, airplane, but also robot manipulator hitting
joint limits
n
Could try by, for example, following formulation: Xt can encode obstacles
n
Or, with constraints, (which would require using an infeasible method):
n
Can work surprisingly well, but for more complicated problems with longer horizons, often get stuck in local maxima that don’t reach the goal
n Helicopter path planning n Swinging up cart-pole n Acrobot
n Configuration Space n Probabilistic Roadmap
n Boundary Value Problem n Sampling n Collision checking
n Rapidly-exploring Random Trees (RRTs) n Smoothing
n obstacles à configuration space obstacles
(2 DOF: translation only, no rotation)
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Configurations are sampled by picking coordinates at random
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Configurations are sampled by picking coordinates at random
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Sampled configurations are tested for collision
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The collision-free configurations are retained as milestones
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Each milestone is linked by straight paths to its nearest neighbors
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Each milestone is linked by straight paths to its nearest neighbors
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The collision-free links are retained as local paths to form the PRM
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The start and goal configurations are included as milestones
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The PRM is searched for a path from s to g
n Initialize set of points with xS and xG n Randomly sample points in configuration space n Connect nearby points if they can be reached from each
n Find path from xS to xG in the graph
n alternatively: keep track of connected components
Typically solved without collision checking; later verified if valid by collision checking
n How to sample uniformly at random from [0,1]n ?
n Sample uniformly at random from [0,1] for each
n How to sample uniformly at random from the surface of the
n Sample from n-D Gaussian à isotropic; then just
n How to sample uniformly at random for orientations in 3-D?
n Pro:
n Probabilistically complete: i.e., with probability one, if run
n Cons:
n Required to solve 2 point boundary value problem n Build graph over state space but no particular focus on
n Basic idea:
n Build up a tree through generating “next states” in the
n However: not exactly above to ensure good coverage
n Finding (approximate) nearest neighbor efficiently
n KD Trees data structure (upto 20-D) [e.g., FLANN] n Locality Sensitive Hashing
n Shortcutting n Nonlinear optimization for optimal control